Library UniMath.CategoryTheory.DisplayedCats.Constructions.FullSubcategory

Require Import UniMath.Foundations.Sets.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Prelude.

Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Functors.
Require Import UniMath.CategoryTheory.DisplayedCats.Total.
Require Import UniMath.CategoryTheory.DisplayedCats.Isos.
Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.

Local Open Scope cat.

1. The construction of the full subcategory as a displayed category

Definition disp_full_sub_ob_mor (C : precategory_ob_mor) (P : C → UU)
  : disp_cat_ob_mor C
  := (P ,, (λ a b aa bb f , unit )).

Definition disp_full_sub_id_comp (C : precategory_data) (P : C → UU)
  : disp_cat_id_comp C (disp_full_sub_ob_mor C P).
Proof.
  split; intros; apply tt.
Defined.

Definition disp_full_sub_data (C : precategory_data) (P : C → UU)
  : disp_cat_data C
  := disp_full_sub_ob_mor C P ,, disp_full_sub_id_comp C P.

Lemma disp_full_sub_locally_prop (C : category) (P : C → UU) :
  locally_propositional (disp_full_sub_data C P).
Proof.
  intro; intros; apply isapropunit.
Qed.

Definition disp_full_sub (C : category) (P : C → UU)
  : disp_cat C.
Proof.
  use make_disp_cat_locally_prop.
  - exact (disp_full_sub_data C P).
  - apply disp_full_sub_locally_prop.
Defined.

2. Univalence

Lemma disp_full_sub_univalent (C : category) (P : C → UU) :
  (∏ x : C , isaprop (P x)) →
  is_univalent_disp (disp_full_sub C P).
Proof.
  intro HP.
  apply is_univalent_disp_from_fibers.
  intros x xx xx'. cbn in *.
  apply isweqcontrprop. apply HP.
  apply isofhleveltotal2.
  - apply isapropunit.
  - intro. apply (@isaprop_is_z_iso_disp _ (disp_full_sub C P)).
Defined.

3. Shortcuts for just the resulting total category

Definition full_subcat (C : category) (P : C → UU) : category := total_category (disp_full_sub C P).

Definition full_subcat_pr1_fully_faithful
  (C : category)
  (P : C → UU)
  : fully_faithful (pr1_category (disp_full_sub C P))
  := fully_faithful_pr1_category _ (λ (a b : full_subcat _ _) _, iscontrunit).

Definition is_univalent_full_subcat (C : category) (univC : is_univalent C) (P : C → UU) :
  (∏ x : C , isaprop (P x)) → is_univalent (full_subcat C P).
Proof.
  intro H.
  apply is_univalent_total_category.
  - exact univC.
  - exact (disp_full_sub_univalent _ _ H).
Defined.

4. The truncation functor

Section TruncationFunctor.

  Context {C : category}.
  Context (P : C → UU).

  Definition disp_truncation_functor
    : disp_functor
      (functor_identity C)
      (disp_full_sub C P)
      (disp_full_sub C (λ X , ∥ P X ∥)).
  Proof.
    use tpair.
    - use tpair.
      + exact (λ X , hinhpr).
      + exact (λ X Y HX HY f , idfun unit).
    - abstract easy.
  Defined.

  Definition truncation_functor
    : full_subcat C P ⟶ full_subcat C (λ X , ∥ P X ∥)
    := total_functor disp_truncation_functor.

  Lemma truncation_functor_fully_faithful
    : fully_faithful truncation_functor.
  Proof.
    intros X Y.
    apply idisweq.
  Defined.

  Lemma truncation_functor_essentially_surjective
    : essentially_surjective truncation_functor.
  Proof.
    intro X.
    refine (hinhfun _ (pr2 X)).
    intro HX.
    exists (pr1 X ,, HX).
    apply (weq_ff_functor_on_z_iso (full_subcat_pr1_fully_faithful _ _)).
    apply identity_z_iso.
  Defined.

End TruncationFunctor.